Problem: Factor the following expression: $5$ $x^2$ $-13$ $x+$ $6$
This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(5)}{(6)} &=& 30 \\ {a} + {b} &=& & & {-13} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $30$ and add them together. The factors that add up to ${-13}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-3}$ and ${b}$ is ${-10}$ $ \begin{eqnarray} {ab} &=& ({-3})({-10}) &=& 30 \\ {a} + {b} &=& {-3} + {-10} &=& -13 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {5}x^2 {-3}x {-10}x +{6} $ Group the terms so that there is a common factor in each group: $ ({5}x^2 {-3}x) + ({-10}x +{6}) $ Factor out the common factors: $ x(5x - 3) - 2(5x - 3) $ Notice how $(5x - 3)$ has become a common factor. Factor this out to find the answer. $(5x - 3)(x - 2)$